[數學普及] Measure Theory (測度論) 簡介 ver. 2

我覺得識當時可以少少technical, 比人知道 technical 究竟咩一回事，真係睇書果時冇咁易嚇親

而家係太technical, 想減少d technical的內容, 但係咁家咁洗濕左個頭好麻煩 *facepalm*
同埋連登講數真係好唔方便, 格式又難搞, 所以整緊個blog

s(x) 最多有 n 個 value (assume A_i disjoint), 所以係散開唔會連續的
X 的 power set 係指 X 的所有 subset
s^{-1} (A) 只係 X 的一個subset, 所以係 power set of X 入面的一粒element only.

oh i get what you mean

s^{-1} (A) 係講緊有咩point in the domain of s 會比 s map左落A度, 所以就算 A 有D 點根本唔會有 domain 的野map落去都無所謂, 當果D點隱形就Ok

e.g. f: {1, 2} -> {3, 4, 5} defined by
f(1) = 3
f(2) = 4

我地依然可以講 f^{-1} ({4, 5}) = {2}, 當個 5 隱形就可以

s( A_1 U A_2) = {c_1, c_2}

如果括號入面果舊野係set, 例如s(A), A係set, 唔係number, 係指緊A 入面ge野會map左去咩set度，即係將所有 s(x) 收集起，而d x係a入面ge點

呢d都普及 不如講下hairy ball theory，Banach–Tarski paradox仲多人有性趣啦....

riemann hypothesis好似正D喎可

呢個點都好過measure theory

七大問題部分都普及既....

當我想個post沉的時候你同我推番出黎

sorry lor 見到有人quote呢個post

唔推囉

edelschwarz

(笑)

辛苦師兄, 不過呢個post冇講topology, 好難教得好measure theory, 就算只係Lebesgue integration.

其實唔講得深入的話，唔需要topology
E.g. Radon measure, Riesz representation theorem, 呢d要識太多野

關於Measure theory定Lebesgue integration, 小弟個人睇法係點都要go through Borel-Sigma-algebra, 至少到 R^n cas

師兄, 冇呢條Theorem, Riemann integrable function on unbounded interval is not necessarily Lebesgue integrable. (我記得做𡃁仔讀書時, Rudin本書有題exercise 係咁)

In other words, Lebesgue integration does not completely generalize Riemann integration.

riemann integral個definition好似唔包improper integral？

師兄你啱, 係當extend Riemann integral去unbounded interval時先fail. 抱歉年代久遠記錯Rudin exercise嗰題嘅意思.

https://math.stackexchange.com/questions/2293902/functions-that-are-riemann-integrable-but-not-lebesgue-integrable/2293956

Btw 其實 lebesgue measure 係 riemann integral ge riesz measure

Yes, 所以我先問師兄不如講得general d, 咁去Riesz representation theorem 明時個效果先更fruitful, 因為Riemann 同 Lebesgue 嘅關係都只係Riesz嘅corollary, 要了解個big picture點都要d basic topology.

尤其個 1-1 correspondence between states on C_C(X) and regular Borel probability measures on X 直落 functional analysis, C*-algebra, 同quantum information theory 都係好 influential 嘅 results.